AP Statistics hypothesis testing, step by step

By Ashraf Bhura, Co-Founder, STEM-Tutor · Updated October 2026

Significance tests are a large part of AP Statistics. A clear, repeatable method makes them much easier.

What a significance test asks

A significance test asks whether sample data give convincing evidence against a claim about a population parameter. The claim is the null hypothesis.

The four steps

1. State the parameter, the hypotheses H₀ and H_a, and the significance level α.
2. Plan: name the test and check the conditions: random, independent (10% condition) and normal or large counts.
3. Do: calculate the test statistic and the p-value.
4. Conclude: compare the p-value with α and answer in context.

Understanding the p-value

The p-value is the probability of getting a result at least as extreme as the one observed, assuming the null hypothesis is true. A small p-value is evidence against the null hypothesis.

Writing the conclusion

For example: "Because the p-value of 0.012 is less than α = 0.05, we reject H₀. There is convincing evidence that the true proportion of students who…" Never say you have proved or accepted the null hypothesis.

Type I and Type II errors

• Type I: rejecting a null hypothesis that is actually true. Its probability is α.
• Type II: failing to reject a null hypothesis that is false.
• Power is the probability of correctly rejecting a false null hypothesis. Larger samples increase power.

Which test?

• One proportion or two proportions: z tests.
• One mean, two means or paired data: t tests.
• Counts in categories: chi-square tests.
• Regression slope: a t test for the slope.

Worked examples

Example 1: a one-sample t test

A machine is meant to fill bottles with 500 ml. A random sample of 25 bottles has mean 510 ml and standard deviation 20 ml. Is the mean different from 500 ml? Use α = 0.05.

• H₀: μ = 500; H_a: μ ≠ 500.
• Conditions: random sample; 25 bottles is less than 10% of production; check the sample data for strong skew or outliers.
• t = (510 − 500)/(20/√25) = 10/4 = 2.5, with 24 degrees of freedom. Two-sided p-value ≈ 0.02.
• Because 0.02 < 0.05, reject H₀. There is convincing evidence that the mean fill differs from 500 ml.

Example 2: a two-proportion z test

60% of 200 students at school A and 50% of 200 at school B say they read daily. Is there a difference? Use α = 0.05.

• Pooled proportion = (120 + 100)/400 = 0.55.
• Standard error = √(0.55 × 0.45 × (1/200 + 1/200)) ≈ 0.0497.
• z = 0.10/0.0497 ≈ 2.01; two-sided p-value ≈ 0.044.
• Because 0.044 < 0.05, there is convincing evidence that the proportions differ.

Errors and power, in context

Suppose H₀ says a new teaching method does not improve scores.

• Type I error: concluding the method improves scores when it does not. The school might adopt a useless method.
• Type II error: concluding there is no improvement when there really is. Students miss out on a better method.
• Increasing power: use a larger sample, increase α (at the cost of more Type I errors), or study a larger true effect.

Common mistakes

• Writing hypotheses about the sample statistic instead of the population parameter.
• Saying the p-value is the probability that H₀ is true.
• Accepting H₀ instead of failing to reject it.
• Skipping conditions or stating them without checking.

Practice questions

Try each one before opening the answer.

Key takeaways

• Hypotheses are about population parameters.
• Check conditions before calculating.
• The p-value assumes H₀ is true.
• Conclude in context, without claiming proof.
• Understand Type I and Type II errors and power.

Where to go next

Review the formulas and the rest of the course.

• AP Statistics formula sheet, explained
• AP Statistics review and study guide
• AP Statistics units, explained

The logic behind significance tests

A significance test works like a courtroom. The null hypothesis is the claim of "nothing unusual": no difference, no effect, the coin is fair. You assume it is true and ask how surprising your data would be in that world. If the data would be very surprising, you have evidence against the null hypothesis. If not, you do not have enough evidence to reject it, which is different from proving it true.

The p-value measures that surprise. A small p-value means data like yours would rarely happen if the null hypothesis were true. The significance level α is the line you draw in advance: how rare is rare enough to reject?

Choosing one-sided or two-sided

Use a one-sided alternative only when the question is about a specific direction, such as "greater than". Use a two-sided alternative when it asks about any difference. Decide before looking at the data; choosing after you see the results is not fair.

Tests and intervals together

A two-sided test at α = 0.05 and a 95% confidence interval agree. If the hypothesised value lies outside the interval, the test rejects it; if it lies inside, the test does not. Intervals add useful information: they show the size of the effect, not just whether one exists.

Frequently asked questions

Is a smaller p-value always more important?

Not necessarily. With very large samples, tiny and unimportant differences can be statistically significant. Always consider the size of the effect in context.

What significance level should I use?

Use the one given in the question. If none is given, 0.05 is the usual choice; state it.

More questions students ask

Can I use a calculator for the test?

Yes, but show the procedure name, the test statistic and the p-value in your answer, not just the calculator result.

What if the conditions are not met?

Say so. The results of the test may not be reliable, and that observation itself can earn credit.

Quick check

Study tip

Write the four steps (state, plan, do, conclude) at the top of every practice question until the routine is automatic.

A final word

Significance tests reward calm, careful thinking. Read the question twice, decide the parameter and the procedure, check the conditions, and finish with a clear sentence in context. With practice, this routine becomes quick and reliable under exam conditions.